Hydrodynamic Traffic Flow Models and Its Application to Studying Traffic Control Effectiveness

Nickolay Smirnov, Alexey Kiselev, Valeriy WSEAS TRANSACTIONS on FLUID MECHANICS Nikitin, Mikhail Silnikov, Anna Manenkova Hydrodynamic traffic flow models and its application to studying traffic control effectiveness Nickolay Smirnov1,2, Alexey Kiselev 2, Valeriy Nikitin1,2, Mikhail Silnikov3, Anna Manenkova2 1 Scientific Research Institute for System Studies of the Russian Academy of Sciences, 36-1 Nakhimovskiy pr., Moscow 117218, RUSSIA 2Moscow M.V. Lomonosov State University, Leninskie Gory, 1, Moscow 119992, RUSSIA, 3 Saint Petersburg State Polytechnical University, 29 Politechnicheskaya Str., 195251 St. Petersburg RUSSIA [email protected] Abstract:-The present research was motivated by the arising difficulties in rational traffic organization the Moscow Government was faced in the recent years. The decrease of the roads handling capacity in the centre of the city (the Boulevard ring, the Gardens ring) caused by the increase of traffic density brought to necessity of re-organizing the traffic flows and constructing of the third ring road around the centre of the city. The present paper contains the results of investigations on mathematical modeling of essentially unsteady-state traffic flows on ring roads. The developed mathematical model is not limited only by the substance conservation equation and empirical relationships between density and velocity, but contains both the continuity (substance conservation) and momentum equations. It presents a closed form mathematical model and suggests the procedure to develop the model parameters based on technical characteristics of the vehicles. Thus, the available experimental data on the traffic flows observations is used only for validation of the model. Key-Words:- Traffic, flow, model, continuous, handling capacity, traffic jam, regulation strategy line of standing cars near the traffic light begins to 1 Introduction grow. Cars approaching from behind come to a stop. The rate of its growth is determined by the First attempts of developing the mathematical mean traffic flux and density on the road. It is, models for traffic flows were undertaken by actually, the velocity of a compression wave Lighthill and Whitham [1], Richards [2], described in [13-18]. The growth of the standing Greenberg [3], Prigogine [4]. A detailed analysis lane is limited by the time of approaching of the of the results obtained could be found in the book rarefaction wave propagating from the traffic light, by Whitham [5]. The authors regarded the traffic when it becomes green. The first characteristics of flow from the position of continuum mechanics this wave moves at a speed of k - the speed of applying empirical relationships between the flow weak disturbances. Then cars start to move slowly density and velocity of vehicles. Development of and the fast cars approaching from behind need to computer technique gave birth to continua models slow down to the low velocity of the traffic flow. application not only for analytical solutions [6] but That gives birth to a traffic jam, which moves also in numerical simulations of traffic flows [7- counter flow. One border of the jam moves at a 10]. Another approach using discrete speed of compression wave, the other is an approximation for public traffic and passengers expanding rarefaction wave. The size of the jam interaction is illustrated by [11-12]. gradually decreases. Depending on the distance of The presence of traffic regulation in the the traffic regulating section from the tunnel, the roads affect essentially flow dynamics. In standing line or traffic jam propagating counter particular, on changing the traffic light for red the flow could enter the tunnel. As it was shown, cars E-ISSN: 2224-347X 178 Volume 9, 2014 Nickolay Smirnov, Alexey Kiselev, Valeriy WSEAS TRANSACTIONS on FLUID MECHANICS Nikitin, Mikhail Silnikov, Anna Manenkova slowing down, the increase of density, then ∂ρ ∂()ρv acceleration bring to an increase of pollutants +=0 emission and decrease of its removal thus ∂∂tx increasing air pollution in the tunnel. (1) The present paper aims at investigating the Then, we derive the equation for the traffic peculiarities of standing lines growth and traffic dynamics. The traffic flow is determined by jams formation in the sections of roads with different factors: drivers reaction on the road regulated traffic. Contrary to the first mathematical situation, drivers activity and vehicles response, models [1-5], the model proposed in [15-18] technical features of the vehicles. The following scoped not only the continuity equation but the assumptions were made in order to develop the differential equation for velocity dynamics which model: takes into account limits for velocity and • It is the average motion of the acceleration of the vehicles, their technical features traffic described and not the motion of the and the peculiarities of the drivers’ reaction on the individual vehicles, that is modelled. road situation. This model has no analogue in the Consequently, the model deals with the mean classical hydrodynamics. The modeled devices of features of the vehicles not accounting for traffic control scope street traffic lights and “laying variety in power, inertia, deceleration way policemen”. length, etc. • The “natural” reaction of all the drivers is assumed. For example, if a driver 2 Autoroute traffic flow model sees red lights, or a velocity limitation sign, or a traffic jam ahead, he is expected to We consider a uni-directional traffic flow. decelerate until full stop or until reaching a Crossings and presence of traffic lights should be safe velocity, and not to keep accelerating with described by specific boundary conditions. We further emergency braking. introduce the Euler’s co-ordinate system with the It is assumed that the drivers are loyal to Ox axis directed along the autoroute and time • the traffic rules. In particular, they accept the denoted by t . The average flow density ρ ()х,t is velocity limitation regime and try to maintain the defined as the relation of the surface of the road safe distance depending on velocity. occupied by vehicles to the total surface of the The velocity equation is then written as road considered: follows: S hnll n tr , dv ρ == = = a; ShLL dt where h is the lane width, L is the sample −+ road length, l is an average vehicle’s length plus a aa=max{− ;min{ a ;a′}}; (2) minimal distance between jammed vehicles, n is Y V ()ρ − v the number of vehicles on the road. With this a′ =+σσω a 1,− y a t x++ y dy 00ρρ()())∫ ( definition, the density is dimensionless changing 0 τ from zero to unit. k 2 ∂ρ a =− . The flow velocity denoted vv(,)= xt can ρ ρ ∂x 0 0 Here, a is the acceleration of the traffic vary from zero to vmax , where vmax is maximal + − permitted road velocity out of the traffic control flow; a is the maximal positive acceleration, a devices zones. From definitions, it follows that the is the emergency braking deceleration; a + and a − maximal density ρ =1 relates to the case when are positive parameters which are determined by vehicles stay bumper to bumper. It is naturally to technical features of the vehicle. The parameter assume that the traffic jam with v0= will take k > 0 is the small disturbances propagation place in this case. velocity (“sound velocity”), as it was shown in [16-17]. The parameter τ is the delay time which Determining the “mass” distributed on a depends on the finite time of a driver’s reaction on road sample of the length L as: the road situation and the vehicle’s response. This L parameter is responsible for the drivers tendency to mdx, = ∫ ρ keep the vehicles velocity as close as possible to 0 the safe velocity depending on the traffic density one can develop a “mass conservation law” in the V () [15-17]: form of continuity equation: ρ E-ISSN: 2224-347X 179 Volume 9, 2014 Nickolay Smirnov, Alexey Kiselev, Valeriy WSEAS TRANSACTIONS on FLUID MECHANICS Nikitin, Mikhail Silnikov, Anna Manenkova −1 0 0 ⎧−k lnρ , v< vmax , ρ =+1v/X l , V ()ρ = ⎨ *m( ()ax) v,00 vv≥ . ⎩ max max kX=+vln101− v 0 /l . The velocity V ()ρ is determined from the max ( ()max ) 0 dependence of the traffic velocity on density in the At vmax = 80 kph the deceleration way of “plane wave” when the traffic is starting from the an automobile of Lada brand is 45 m, which gives initial conditions ρ0 =1 and v0= , with account k = 35 kph when the mean length of a vehicle is of velocity limitation from above 0 . The 5 m (including the minimal distance between ()vv≤ max motionless vehicles). For this velocity v0 , the value of τ could be different for the cases of max acceleration or deceleration to the safe velocity maximal safe density is ρ* = 0.1. The maximal V ()ρ : accelerations of an automobile of this type are 2 2 + + − ⎪⎧τρ, V() < v a = 1.63 m/s and a = 5.5 m/s . τ = ⎨ − The “sound velocity” k estimated as above ⎩⎪τρ, V() ≥ v fits the experimental data [2, 3, 5] rather good. The remaining parameters in (2) are the Summarising the above, one comes to the following. YYLx=min{ 0 , −} is the model consisting of two PDE’s (1) and (2) characteristic visibility ahead depending on the describing the automobile traffic dynamics on a weather, ω ()y is the “weight” (in the single-lane road. The divergent form of those equations is: mathematical sense) of the traffic ahead of the ∂ρ ∂()ρv vehicle, which influences on the drivers decision +=0, of changing the velocity and which could be ∂∂tx determined as follows, for example: 2 ∂ ()ρv ∂ ()ρv ω0 ()y ⎧1, 0 ≤≤yY0 +=ρa , ωω()y ==Y , 0 ⎨ , ∂∂tx 0, yyY<>0, ω y dy ⎩ 0 and the acceleration a is determined by three last ∫ 0 () expressions in (2).

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